Geodesic
Planar maps, circle patterns and 2D gravity
Annales de l’Institut Henri Poincaré D, Tome 1 (2014) no. 2, pp. 139-183.

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Via circle pattern techniques, random planar triangulations (with angle variables) are mapped onto Delaunay triangulations in the complex plane. The uniform measure on triangulations is mapped onto a conformally invariant spatial point process. We show that this measure can be expressed as: (1) a sum over 3-spanning-trees partitions of the edges of the Delaunay triangulations; (2) the volume form of a Kähler metric over the space of Delaunay triangulations, whose prepotential has a simple formulation in term of ideal tessellations of the 3d hyperbolic space ℍ 3 ; (3) a discretized version (involving finite difference complex derivative operators ∇,∇ ¯) of Polyakov's conformal Fadeev-Popov determinant in 2d gravity; (4) a combination of Chern classes, thus also establishing a link with topological 2d gravity.

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DOI : 10.4171/aihpd/5
Classification : 05-XX, 52-XX, 60-XX, 81-XX
Keywords: Circle pattern, Random maps, Conformal invariance, Kähler geometry, 2D gravity, topological gravity 
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     title = {Planar maps, circle patterns and {2D} gravity},
     journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
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David, François; Eynard, Bertrand. Planar maps, circle patterns and 2D gravity. Annales de l’Institut Henri Poincaré D, Tome 1 (2014) no. 2, pp. 139-183. doi : 10.4171/aihpd/5. http://geodesic.mathdoc.fr/articles/10.4171/aihpd/5/

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